Angles of elevation and depression, and bearings with trigonometry

Right-angled trigonometry solves real problems: the height of a tree, the distance to a boat, or how far east a ship has sailed.

Practise this lesson free (6 questions)

Elevation and depression

The angle of elevation is measured up from the horizontal.
The angle of depression is measured down from the horizontal.
The angle of depression from A to B equals the angle of elevation from B to A (alternate angles).
22.6° angle of depression 22.6° angle of elevation boat 50 m 120 m The two angles are equal (alternate angles).
Angle of depression from the cliff top = angle of elevation from the boat (alternate angles).

Example

You stand 20 m from a tree. The angle of elevation of the top is 35°.
tan 35° = h20, so h = 20 tan 35° = 14.0 m.
A cliff is 50 m high and a boat is 120 m from its foot. Angle of depression = tan−1(50120) = 22.6°.
35° 20 m h tan 35° = h ÷ 20, so h = 14.0 m
Height of a tree: h = 20 × tan 35° = 14.0 m.

Bearings and trigonometry

A ship sails 8 km on a bearing of 060°. Draw North, then the 60° angle.
East: 8 sin 60° = 6.93 km. North: 8 cos 60° = 4 km.
B is 5 km east and 12 km north of A: angle from North = tan−1(512) = 22.6°, so the bearing is 022.6°.
N 60° 8 km east 6.93 km north4 km A
Bearing 060°, 8 km: east 8 sin 60° = 6.93 km, north 8 cos 60° = 4 km.
Worked example

From 20 m away, the angle of elevation of the top of a tree is 35°. Find its height.

  1. tan 35° = h20
  2. h = 20 × tan 35° = 14.004...

Answer: 14.0 m

Key idea

Elevation is up from the horizontal, depression is down. Draw the right-angled triangle, label O, A and H, then use SOHCAHTOA. For bearings, start from North.

Key words
Alternate angles
Angles on opposite sides of a line that crosses two parallel lines, inside the parallel lines. They are equal (a Z shape).
Bearing
A direction measured clockwise from north, written with three figures, such as 045°.

All GCSE and International GCSE Mathematics key terms and definitions →

Quick check

Try two quick questions

See if it has sunk in. Answers are marked straight away.

Extra practice

One more question

Try each one, then tap to check.

  1. From the top of a cliff, the angle of depression of a boat is 25°. What is the angle of elevation of the cliff top from the boat?

    1. 25°
    2. 65°
    3. 155°
    4. 50°
    Show the answer

    Answer: 25°

    The two angles are alternate angles between parallel horizontal lines, so both are 25°.

Check you have got it

Answer 6 quick questions with instant marking. If you get one wrong, GCSE-ready shows you why and gives you another go. It is free, and you do not need an account.

Practise this lesson free (6 questions)All GCSE and International GCSE Mathematics lessons

Video for this lesson coming soon. Subscribe on YouTube

Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

Related lessons